288 ANALYTICAL MECHANICS an acceleration - 7 -r, while the earth imparts to the sui r* an acceleration equal to 7 — • Suppose at any instant we impart to both the sun anc the earth a velocity equal and opposite to that of the sur fW\ and apply an acceleration —j. This will bring the sun t< rest and keep it at rest, without altering the motion of th< earth relative to the sun.* This reduces the problem o: the motion of the earth to that of a particle moving in i central field of force where the acceleration is ,, N M m T" or / (r) = •£ (VII and [F=-M2J, (VIII rz where M = 7 (-W + m)- Substituting from equation (VIII) in equation (V) w< obtain for the equation of the orbit. Let uf = u — j~, then th( equation of the orbit takes the form t'=0. (2; (to2 ' In order to integrate equation (2) let t; = --—, * The acceleration of a particle relative to another moving particle i found by adding the negative of the acceleration of the second particle t< that of the first.